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Why a negative times a negative makes sense

Use the repeated addition model of multiplication to give an understanding of multiplying negative numbers. Created by Sal Khan.

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Video transcript

So you, as the ancient philosopher in mathematics have concluded in order for the multiplication of positive and negative numbers to be consistent with everything you've been constructing so far with all the other properties of multiplication that you know so far that you need a negative number times a positive number or a positive times a negative to give you a negative number and a negative times a negative to give you a positive number and so you accept it's all consistent so far.. this deal does not make complete concrete sense to you, you want to have a slightly deeper institution than just having to accept its consistent with the distributive property and whatever else and so you try another thought experiment, you say "well what is just a basic multiplication way of doing it?" So if I say, two times three, one way to to conceptualize is basic multiplication is really repeating addition, so you could view this as two threes so let me write three plus three and notice there are two of them, there are two of these or you could view this as three twos, and so this is the same thing as two plus two plus two and there are three of them, and either way you can conceptualize as you get the same exact answer. This is going to be equal to six, fair enough! Now, you knew this before you even tried to tackle negative numbers. Now let's try to make one of these negatives and see what happens. Let's do two times negative three, I want to make the negative into a different color. Two times negative three. Well, one way you could view this is the same analogy here, it's negative three twice so it would be negative.. I'll try to color code it negative three and then another negative three or you could say negative three minus three or, and this is the interesting thing, instead of over here there's a two times positive three you added two, three times. But since here is two times negative three you could also imagine you are going to subtract two, three times So instead of up here, I could written two plus two plus two because this is a positive two right over here, but since we're doing this over negative three we could imagine subtracting two, three times, so this would be subtracting two (repeated) subtract another two right over here, subtract another two and then you subtract another two notice you did it, once again, you did it three times, so this is a negative three, so essentially you are subtracting two, three times. And either way, you can conceptualize right over here, you are going to get negative six negative six is the answer. Now, so you are already starting to feel better about this part right over here negative times a positive, or a positive times a negative is going to give you a negative. Now lets take to the really un-intuitive one and measure negative times a negative, and all of a sudden negatives kind of cancel to give you a positive. Now why is that the case? Well we can just build from this example right over here. Let's say we had a negative two, lets say we had negative two, let me do it a different color, let's say we had a negative two, I already used this color negative two times negative three. So now, we can d- actually I'll do this one first. Let's do multiplying something by negative three so we'll repeatedly subtract that thing three times whatever that thing is so now the thing isn't a positive two so the thing over here is a positive two but the thing we're going to subtract is a negative two So let me make it clear, this says we are going to subtract something three times, so we subtract something three times, so subtracting something (repeatedly) three times That's what this part right over here tells us and we'll do this, exactly three times Over here, it was a positive two we subtracted three times, now we're going to do a negative two, now we're going to do a negative two and we know from subtracting negative numbers, we already built this intuition that subtracting a negative is the same thing it's the same thing as adding a positive, and so this this is going to be the same thing as two plus two plus two and we're told once again, gives you a positive six, you can same use the same logic over here, now instead of adding negative three twice, really I could have written this as negative three as this example negative three negative three, and we added it we added it, now let me put a plus here to make it clear over here we added it twice, we added negative three two times, or here since we have a negative two, we're going to subtract to negative three twice, so we're going to subtract something and we're going to subtract something again, and that something is going to be our negative three, it's going to be our negative three, so negative, negative and put our three right over here and once again, subtracting negative three is like taking away someone's debt, which is essentially giving them money, this is the same thing as adding three plus three which is once again six. So now you, the ancient philosopher, feel pretty good. Not only this all consistent with all the mathematics you know the distributive property is also the property of multiplying something times something all these things you already know, and now this actually makes conceptual sense to you, this is actually very consistent with with your notion, your original notion, or one of the possible notions of multiplication which is as repeated addition